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Fermat's Last Theorem: Infinite Descent

We dived in the blue Everyone at infinite descent is great! So helpful and knowledgeable! The prices were reasonable and it was an overall great experience for all of us! Infinite descent is the only dive shop docking right onto the gulf. The staff are really knowledgeable and helpful. Maria and Eric are especially great! The dive trips either fresh or salt water are reasonably priced and would recommend for beginners or experienced divers!

Outstanding 2 tank dive experience. All the staff was amazing. Maria handled our booking and was very responsive.

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Answered our email in minutes and called us back. Not the norm in the dive shop world! Great day! This community of people is very friendly and speak to you in a way that helps you better your diving skills.

Proof by infinite descent

Not only are they great teachers but Eric, the GM, spoke to myself and the rest of the group like we were longtime friends and made us feel extremely comfortable in water conditions that we would not have otherwise felt comfortable in. OH and checkout their beach bar a few doors down! Just as awesome as the shop! An absolutely wonderfilled dive experience with the ID team.

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Sea Turtle Sunday after Black Friday is a hit! We visited this place as it was the closest dive shop to where we were staying, we wanted to dive over the weekend but couldn't due to bad conditions, the guys at the shop recommended the springs further up north and gave us some tips. We hired equipment from here all was of too quality, and the service was excellent.

Nate and Connor were cool guys and gave us tips on where to go and what to see. The shop also has a bar next door which is great where we met further members of the dive crew will and memory who also talked about diving and trios the group were taking next year.

Great to meet such an awesome group who are passionate about what they do and the service they deliver. We hope to be back soon and catch some diving with these guys.

What Is Infinite Descent?

Highly recommend to anyone who is experienced or wants to get involved in diving and pop by the dive bar for some good drinks and chat. Madeira Beach. Profile JOIN. Log in to get trip updates and message other travelers. Awesome freshwater dive experience - Infinite Descent.

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  • Beast Academy. Engaging math books and online learning for students ages Books for Ages Beast Academy Online. AoPS Academy. Nationwide learning centers for students in grades Find a Campus. Sign In. Forums Contests Search Help. AoPS Wiki. Resources Aops Wiki Infinite Descent. Thus, the essential technique of mathematical induction consists of establishing the P-status of one or more element s e[k] by prior means, and then invoking some general relation between the P-status of one element and the P-status of some others in the ordered sequence of elements.

    Notice that we haven't used any notion of the relative "magnitudes" of the elements e[] they may not even have magnitudes. We have simply used the ordering and completeness of the sequence. Now let me describe my conception of "infinite descent", which seems to be different than some popular conceptions. We have in general an infinite set of elements, each of which has a finite non-negative "magnitude" norm , and the difference between any two distinct magnitudes is greater than some fixed finite quantity D.

    Notice that these elements need not be given in any particular linear order. For example, we might have elements e. Suppose we can show that if any one of these elements has the property P, then the property P must also be possessed by another element whose magnitude is strictly less than the magnitude of the former. Clearly this is impossible, because it implies an infinite descent in steps no smaller than D, beginning from a finite starting magnitude, whereas there are only finitely many distinct magnitudes less than any given magnitude. It might seem this argument can be re-cast in the form of a simple induction on the linear sequence of magnitudes, but it can't.

    To see why, suppose we arrange the elements of our set according to their magnitudes, and we get something like this. To apply induction here, we would first need to regard all the elements of a certain magnitude as "equivalent" with respect to the property P i.

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    This is already a slight problem, since many properties depend on more than just the norm of an element. However, we can probably fix that up. Fair enough, but this doesn't prove that m[k] cannot possess -P. However, suppose we cannot determine by independent means whether m[0], or any other specific magnitude, possesses P. Does our induction argument whether we phrase it in positive or contra-positive form enable us to conclude anything now? Evidently not, because it relies on knowledge of the P-status of some specific element to make it effective.

    Now suppose we're given one additional piece of information: If m[k] possesses P, then there is a magnitude m[j] strictly less than m[k] such that m[j] possesses P.

    Formal Definition

    This information still does not enable us to use our induction argument positive or contra-positive to give a definite result, because we still haven't been given the P-status of any magnitude. However, we can now invoke an infinite descent to prove that no magnitude can possess P. We reach this conclusion simply from the fact that there are only a finite number of distinct magnitudes less than any given magnitude, whereas we have shown that if m[k] possesses P there must be infinitely many distinct magnitudes less than m[k].

    This argument is actually much closer to the pigeon-hole principle than to induction. In fact, one could argue that it is the pigeon-hole principle. In summary, induction whether applied upward or downward, and in the positive or contra-positive sense always relies on the known P-status of at least one element, whereas the conclusion in an infinite descent is based purely on the structure of the implicative relation and the field of available elements.

    Thus, they clearly operate on a fundamentally different principle. One could make a strong case that infinite descent is equivalent to a specialized application of the pigeon-hole principle which is another high-order principle , but not to induction. One more comment on the above schema for descent :.